I was in a band at one point that has a verified Facebook page. I'm not exactly sure how it works with IG but I would assume that it's pretty similar to FB since it's the same company. On FB you could go two different routes: one is to submit a request to FB with photo ID verifying you as a legitimate representative of the account (we didn't have to do it this way so I'm not sure how well it works for accounts that aren't really famous), the second route is to have a company which is affiliated with other existing "verified" pages (i.e. management or PR company) to submit the request for you. These seem to work the most often because I know several other bands who have gotten verified this way once they sign on to a particular roster.
Regardless, it's hilarious that someone would say that this will cost you $10k because you could just as easily work out a deal with a PR agency (depending on your niche, I guess) to submit the request for you "under the table" for a fraction of that price without signing on for a full-term contract.